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SCHOOL OF CIVIL ENGINEERING

RESEARCH REPORT R931FEBRUARY 2013

ISSN 1833-2781

SHEAR BUCKLING OF CHANNEL SECTIONS WITH SIMPLY SUPPORTED ENDS USING THE SEMI-ANALYTICAL FINITE STRIP METHOD

CAO HUNG PHAMGREGORY J. HANCOCK

SCHOOL OF CIVIL ENGINEERING

SHEAR BUCKLING OF CHANNEL SECTIONS WITH SIMPLY SUPPORTED ENDS USING THE SEMI-ANALYTICAL FINITE STRIP METHOD RESEARCH REPORT R931 GREGORY J. HANCOCK CAO HUNG PHAM February 2013 ISSN 1833-2781

Shear Buckling of Channel Sections with Simply Supported Ends using the Semi-Analytical Finite Strip Method

School of Civil Engineering Research Report R931 Page 2 The University of Sydney

Copyright Notice School of Civil Engineering, Research Report R931


Gregory J. Hancock Cao Hung Pham February 2013 ISSN 1833-2781 This publication may be redistributed freely in its entirety and in its original form without the consent of the copyright owner. Use of material contained in this publication in any other published works must be appropriately referenced, and, if necessary, permission sought from the author. Published by: School of Civil Engineering The University of Sydney Sydney NSW 2006 Australia This report and other Research Reports published by the School of Civil Engineering are available at http://sydney.edu.au/civil



ABSTRACT Buckling of thin-walled sections in pure shear has been recently investigated using the Semi-Analytical Finite Strip Method (SAFSM) to develop the “signature curve” for sections in shear. The method assumes that the buckle is part of an infinitely long section unrestrained against distortion at its ends. For sections restrained at finite lengths by transverse stiffeners or other similar constraints, the Spline Finite Strip Method (SFSM) has been used to determine the elastic buckling loads in pure shear. These loads are higher than those from the SAFSM due to the constraints.

The SFSM requires considerable computation to achieve the buckling loads due to the large numbers of degrees of freedom of the system. In the 1980’s, Anderson and Williams developed a shear buckling analysis for sections in shear where the ends are simply supported based on the exact finite strip method. The current report further develops the SAFSM buckling theory of YK Cheung for sections in pure shear accounting for simply supported ends using the methodology of Anderson and Williams. The theory is applied to the buckling of plates of increasing length and channel sections in pure shear also for increasing length. The method requires increasing numbers of series terms as the sections become longer. Convergence studies with strip subdivision and number of series terms is provided in the report.

KEYWORDS Cold-formed channel sections; Simply supported ends; Shear buckling analysis; Finite strip method; Semi-analytical finite strip method; Complex mathematics.



TABLE OF CONTENTS

ABSTRACT .................................................................................................................................................... 3

KEYWORDS .................................................................................................................................................. 3

TABLE OF CONTENTS ................................................................................................................................. 4

INTRODUCTION ............................................................................................................................................ 5

THEORY ........................................................................................................................................................ 5

Strip Axes and Nodal Line Deformations .................................................................................................... 5

Plate Buckling Deformations with Simply Supported Ends ......................................................................... 6

Forces along The Nodal Lines .................................................................................................................... 6

Total Energy ............................................................................................................................................... 7

Stiffness and Stability Matrices ................................................................................................................... 8

Assembly of Stiffness And Stability Matrices .............................................................................................. 8

EIGENVALUE ROUTINES ............................................................................................................................. 9

Sturm Sequence Property .......................................................................................................................... 9

Direct Computation of Sign Count of ([A] - λ [B]) ........................................................................................ 9

Eigenvector Calculation .............................................................................................................................. 9

Computer Program bfinst8.cpp ................................................................................................................... 9

SOLUTIONS TO PLATES AND SECTIONS IN SHEAR ................................................................................ 9

Plate Simply Supported on Both Longitudinal Edges ................................................................................. 9

Lipped Channel Section in Pure Shear ..................................................................................................... 11

Web-Stiffened Channel in Pure Shear ...................................................................................................... 14

CONCLUSIONS ........................................................................................................................................... 16

ACKNOWLEDGEMENTS ............................................................................................................................ 16

REFERENCES ............................................................................................................................................. 16

Appendix A: Buckling Stress and Shear Buckling Coefficient for Plain Lipped Channel ............................. 18

Appendix B: Buckling Stress and Shear Buckling Coefficient for Stiffened Web Channel .......................... 19

Appendix C: Flexural Stiffness and Stability Matrices For Mth Series Term ................................................ 20

Appendix D: Membrane Stiffness and Stability Matrices For Mth Series Term ............................................ 22

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The strip nodal line flexural deformations in Fig. 1 are given in vector format by:

{δF} = (w1, θx1, w2, θx2)T (1)

Similarly, the strip nodal line membrane deformations in Fig. 1 are given in vector format by:

{δM} = (u1, v1, u2, v2)T (2)

In the finite strip method of Cheung (1968, 1976), longitudinal variations of displacement are described by harmonic functions and transverse variations are described by polynomial functions.

PLATE BUCKLING DEFORMATIONS WITH SIMPLY SUPPORTED ENDS

For the analysis of plates and sections with simply supported ends, the strip nodal line deformations are regrouped into those associated with variation according to the sine function, and those associated with variation according to the cosine function as {δs} = (v1, w1, θx1, v2, w2, θx2)T and {δc} = (u1, u2)T respectively. The deformations {δ} of a strip for μ series terms assuming simply supported ends can therefore be expressed by:

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FORCES ALONG THE NODAL LINES

There are N nodal lines corresponding to the intersection of strips both at plate junctions and within plates subdivided into strips. The forces at the nodal lines can be computed at a given load factor λE from the unrestrained stiffness and stability matrices for the mth series term derived in Hancock and Pham (2011 2012) using:

F H δ (5)

where [Hm] = [Km] – λE [Gm] The stiffness matrix [Km] is real and the stability matrix [Gm] is real when the shear stresses shown in Fig. 2 are zero but complex Hermitian when the shear stresses are non-zero. The load factor λE applies to the stresses shown for each strip as in Fig. 2. Equation 5 for the nth series term can be partitioned according to the sequence in Equation 3 so that:

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The forces are out-of-phase with the displacements so that the multiple i on {Fc} and {δc} accounts for the 90 degree phase difference between these two components when the shear is zero, so that [Hn] is real and symmetric for such cases.

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STIFFNESS AND STABILITY MATRICES

For equilibrium, the theorem of minimum total potential energy with respect to each of the elements {δm} is:

E 0 (10)

The result is:

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where

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The matrices [HmR] and [Qmn], and the vectors {δm}, {δn}, are real.

In Equation 11, when the shear stress τ is zero, Qmn is zero and hence the individual series terms are uncoupled in m = 1, 2, .... , μ and each can be solved independently. Further, [Qmn] is the transpose of [Qnm] since [Hm

I] is Hermitian so that the resulting stiffness and stability matrices are symmetric. For example, when μ = 3 (3 series terms), the full stiffness and stability matrix derived from Equation 11 can be represented as follows:

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0 Q HR

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The matrix [H] has 4 * N * μ degrees of freedom. If the rows and columns in the matrix [H] are organised so that each degree of freedom is taken over the μ series terms, then the half-bandwidth of the matrix is simply μ times the half-bandwidth of the problem with one series term. This speeds the computation of the eigenvalues and eigenvectors considerably.

ASSEMBLY OF STIFFNESS AND STABILITY MATRICES

The component flexural stiffness and stability matrices used to compute [Km] and [Gm] in Eq. 5 (and hence [Hm]) was derived in Hancock and Pham (2011, 2012) to be:

kF λE gF δF 0 (14)

The flexural stiffness matrix [kFm] is real and the flexural stability matrix [gFm] is real if the shear stress τ is zero. However, the stability matrix [gFm] is complex Hermitian if the shear stress is non-zero. They are given for the mth series term in Appendix C. The component membrane stiffness and stability matrices used to compute [Km] and [Gm] in Eq. 5 were derived by Hancock and Pham (2011, 2012) to be:

kM λE gM δM 0 (15)

The membrane stiffness matrix [kM] is real and the membrane stability matrix [gM] is real. They are given for the mth series term in Appendix D. The term λE is the load factor. For folded plate assemblies including thin-walled sections such as channels, (14) and (15) must be transformed to a global co-ordinate system to assemble the stiffness [Km] and stability [Gm] matrices of the folded plate assembly or section for the mth series term.



It is clear that only [gF] has complex terms. So only [H11nI] and [H11m

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EIGENVALUE ROUTINES

STURM SEQUENCE PROPERTY

From the theory of equations (Turnbull, 1946), the leading principal minors of [C]-λ[I] (where [I] is a unit matrix) form a Sturm sequence. The leading principal minor of order r is given by det ([Cr] –λ[I]) where [Cr] is the leading principal sub-matrix of order r of [C]. The first term of the Sturm sequence is the leading principal minor of order r=0 and is defined to be unity. The number of eigenvalues greater than λ is equal to the number of agreements in sign between consecutive members of the Sturm sequence from r = 0 to r = n where n is the dimension of the matrix [C]. This property is very useful in isolating the range of λ in which a particular eigenvalue is located. The eigenvalue corresponding to a particular mode number can be isolated by bisection between values of λ which bound the eigenvalue.

DIRECT COMPUTATION OF SIGN COUNT OF ([A] - λ [B])

Peters and Wilkinson (1969) have shown that the sign of det([Ar] – λ[Br]) is the same as that of det([Cr] – λ[I]). Consequently, it is possible to apply the Sturm sequence directly to ([A] - λ[B]) without the need to transform to the standard eigenvalue problem det([C] - λ[I]) = 0. For the finite strip buckling analysis given by (11), the [G] component of [H] is chosen as [A] and the [K] component of [H] is chosen as [B] so that the computed eigenvalues λ of ([A] - λ[B]) are the reciprocals of the load factors λE

EIGENVECTOR CALCULATION

Wilkinson (1958) has produced a method for computing the eigenvector {δ} of (11) by solving the equations at the value of λE for a unit right hand side vector {1} replacing {0} in (11). The process is usually repeated once to purify the eigenvector with the unit vector {1} replaced by {δ} from the first iteration. This method has been used in the calculations in this report.

COMPUTER PROGRAM bfinst8.cpp

A computer program bfinst8.cpp has been written in Visual Studio C++ to assemble the stiffness and stability matrices given by Equation 11 and to solve for the eigenvalues using the Sturm sequence property described in the Sections above, and to compute the corresponding eigenvectors as per the Section above. The program stores the real [K] matrix and the real [G] and complex [GI] components of the stability matrices in order to extract the eigenvalues and eigenvectors. The results from this analysis are given the notation reSAFSM (restrained SAFSM) in this report to distinguish it from the unrestrained SAFSM in Hancock and Pham (2011, 2012).

SOLUTIONS TO PLATES AND SECTIONS IN SHEAR

PLATE SIMPLY SUPPORTED ON BOTH LONGITUDINAL EDGES



The solution for a plate simply supported along both longitudinal edges determined using the reSAFSM analysis is compared with the classical solution of Timoshenko and Gere (1961) Item 9.7 Buckling of rectangular plates under the action of shearing stresses. The equation for the elastic buckling of a rectangular plate is given (Timoshenko and Gere (1961)) as:

τ D

(17)

where D is the plate flexural rigidity, b is the width of the plate which may consist of multiple strips and kv is the plate buckling coefficient in shear. The analysis is carried out for 8 equal width strips and an increasing number of series terms. Aspect ratios of 1:1, 2:1, 3:1 and 4:1 have been investigated and the solutions for the buckling coefficient kv are compared in Table 1 with those of Timoshenko and Gere Table 9-10. It is clear that the solutions become more accurate with increasing numbers of series terms, and that more terms are required for higher aspect ratios. The solutions converge to values slightly lower than those of Timoshenko and Gere (1961) presumably because they used less series terms. These kv values can be compared with that for a buckle in an infinitely long section of 5.3385 given in Plank and Wittrick (1974) and Hancock and Pham (2011, 2012). Anderson and Williams (1985) achieved values of 9.37, 9.35 and 9.32 for 6 strips and 3, 4 and 6 series terms respectively using the exact strip formulation for a square plate simply supported along all four longitudinal edges.

Aspect ratio Timoshenko and Gere (1961)

3 series terms 4 series terms 6 series terms

1:1 9.34 9.379 9.366 9.332 2:1 6.6 6.691 6.564 6.551 3:1 5.9 6.644 5.898 5.849 4:1 5.7 7.219 6.029 5.645

Table 1 Buckling coefficients kv for simply supported square and rectangular plates

The buckling modes for an aspect ratio of 2:1 and 6 series terms determined from bfinst8.cpp are plotted as a contour plot using Mathcad in Fig. 3. The buckling mode has one elongated local buckle of the web satisfying the simply supported boundary conditions on all four edges. The contour line finishing at the centres of the ends is the zero displacement contour.

Figure 3. Shear Buckling Mode of Plate of Aspect Ratio 2:1 with Four Edges Simply Supported (6 Series Terms)



LIPPED CHANNEL SECTION IN PURE SHEAR

In order to extend the study to lipped channel sections, a 200mm deep lipped channel with flange width 80mm, lip length 20mm and thickness 2mm as studied by Pham and Hancock (2009a, 2012b) has been used. These dimensions are all centreline and not overall. In Pham and Hancock (2009a), three different shear stress distributions have been investigated. These are uniform shear in the web alone (called Cases A/B), uniform shear in the web and flanges (called Case C), and a shear stress equivalent to a shear flow as occurs in a channel section under a shear force parallel with the web through the shear centre (Case D as shown in Fig. 4). In this report, only Case D is studied as it is the most representative of practice. The shear flow distribution is not in equilibrium longitudinally as this can only be achieved by way of a moment gradient in the section. However, it has been used in these studies to isolate the shear from the bending for the purpose of identifying pure shear buckling loads and modes. The finite strip buckling analysis allows the uniform shear stresses in each strip, as shown in Fig. 2, to be used to assemble the stability matrix [kg] of each strip then the system stability matrix [G]. Fig. 4 demonstrates that the shear flow in each strip is uniform. In the studies of plain channels in this report, the web is divided into sixteen equal width strips, the flanges into ten each and the lips into two each making 40 strips and 41 nodal line with a total of 164 degrees of freedom. Adequate accuracy can be achieved for engineering purposes with 18 strips, 19 nodal lines and 76 degrees of freedom. However, 40 strips have been used in this case for accurate benchmarking against the SFSM analyses.

Figure 4. Shear Flow Distribution Assumed (Case D)

The SAFSM and reSAFSM curves of buckling stress versus half-wavelength/length are compared in Fig. 5. The reSAFSM graph (circles) of buckling stress versus length (as opposed to half-wavelength for the SAFSM) was computed using the bfinst8.cpp program described above with 8 series terms. The reSAFSM analysis assumes no cross-section distortion at both ends of the section under analysis (Z = 0, L). For a section of length 200mm, the reSAFSM analysis gives a buckling coefficient kv of 10.017 for the web which is higher than that for a square panel in shear at 9.34 due to the flange restraint. The buckling mode is shown in Fig. 6 and encompasses a single buckle half-wavelength. At L = 600mm, 1000mm and 1600mm, the buckling modes are shown in Fig. 7, 8 and 9 respectively and involve 3, 5 and 6 local buckle half-waves respectively. At L = 2000mm, the buckling mode involves one half-wavelength and is a type of distortional buckle and is shown in Fig. 10. For comparison, the SAFSM graph (squares) of buckling stress versus buckle half-wavelength (signature curve) derived in Hancock and Pham (2011, 2012) using the program bfinst7.cpp is also shown in Fig. 5 for the lipped channel for a range of buckle half-wavelengths from 30mm to 10000mm. The graph reaches a minimum at approximately 200mm half-wavelength then rises and starts to drop at about 800mm. The buckling coefficient kv corresponding to the minimum point is 6.583 based on the average stress in the web (τav = V/Aw) computed from the shear load V on the section divided by the area of the web Aw. This compares with an asymptotic value of 6.60 for multiple local buckles using the reSAFSM analysis.



Figure 5. Buckling Stress versus Length/Half-Wavelength from SAFSM (bfinst7.cpp) and reSAFSM (bfinst8.cpp) for Simple Lipped Channel

Figure 6. Simple Lipped Channel Shear Buckling Mode at L = 200mm


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reSAFSM (bfinst8.cpp)

SAFSM (bfinst7.cpp)








To validate the accuracy of the reSAFSM analysis, the buckling stresses and coefficients are compared in Appendix A with the Spline Finite Strip Method (SFSM) reported in Pham and Hancock (2009a, 2012b). The SFSM values have also been computed for a section with the same strip subdivision of 40 strips and can be regarded as an accurate solution for benchmarking. They were given previously in Hancock and Pham (2011, 2012). In general, the reSAFSM values are higher than the SFSM values by less than 1%. However, at L=1600mm, the error increases to approximately 4% as the use of only 8 series terms does not accurately predict the behaviour when at least 6 buckle half-waves occur as shown in Fig. 9.

WEB-STIFFENED CHANNEL IN PURE SHEAR

Shear buckling of thin-walled channel sections with intermediate web stiffeners have been studied using the SFSM by Pham and Hancock (2009b) and using the SAFSM by Hancock and Pham (2011, 2012) and Pham SH, Pham CH and Hancock (2012a, 2012b). In this report, the particular web stiffener used is the same as that in Hancock and Pham (2011, 2012) and has a rectangular indent of 5mm over a depth of 80mm located symmetrically about the centre of the web. Swage stiffeners of this type are common in practice. The web is divided into 6 equal width strips for each of the 2 outer vertical elements in the web, 8 for the inner vertical element of the web, the stiffeners in the web into 2 each, the flanges into 6 each and the lips into two each making 40 strips and 41 nodal line with a total of 164 degrees of freedom, the same number as for the simple lipped channel. The SAFSM and reSFSM curves of buckling stress versus half-wavelength/length are compared in Fig.11. The effect of the different end conditions (Z = 0,L) between the SAFSM (squares) and reSAFSM (circles) are clear from this comparison. The shape of the two curves is remarkably similar except that the SFSM shows several plateau points at approximately 200mm and 1000mm lengths corresponding to the switch in buckling modes. The differences in the modes at these switch points are shown in Figs. 12, 13 and 14.

Figure 11. Buckling Stress versus Length/Half-Wavelength from SAFSM (bfinst7.cpp) and reSAFSM (bfinst8.cpp) for Stiffened web Channel

0

200

400

600

800

1000

10 100 1000 10000

Max

imum

Stre

ss in

Sec

tion

at B

uckl

ing

(MP

a)

Buckle Half-Wavelength/Length (mm)

reSAFSM (bfinst8.cpp)

SAFSM (bfinst7.cpp)



Figure 12. Stiffened web channel shear buckling mode at L = 200mm






It is interesting to observe that the stiffened web channel has only two buckle half-waves at a length of 1000mm compared with five for the plain channel. The effect of the stiffener is to decrease the number of buckle half-waves with a corresponding increase in the buckling stress. To validate the accuracy of the reSAFSM analysis for the web stiffened channel, the buckling stresses and coefficients are compared in Appendix B with the Spline Finite Strip Method (SFSM) reported in Hancock and Pham (2011). The SFSM values have also been computed for a section with 40 strips and can be regarded as an accurate solution for benchmarking. In general, the reSAFSM values are higher than the SFSM values by less than 1% for lengths greater than 600mm. At shorter lengths, the error is slightly higher decreasing from 3.8% at 100mm to 1.8% at 200mm for the mode shown in Fig. 12 then falling rapidly to 0.5% at 400mm.

CONCLUSIONS

A new version of the semi-analytical finite strip buckling analysis of thin-flat-walled structures under combined loading with simply supported end conditions has been derived and programmed in Visual Studio C++. The method has been called reSAFSM to reflect the restrained ends. The method uses multiple series terms to allow for the increasing numbers of buckle half-waves as the sections become longer. The method includes the extraction of the eigenvalues and eigenvectors from the matrices produced when thin-walled sections are subjected to shear in addition to compression and bending. The method is based on the theory of Anderson and Williams applied to the general complex matrices developed by Plank and Wittrick. The method using up to 6 series terms has been checked against the solutions of Timoshenko and Gere for simply supported rectangular plate in pure shear and found to produce accurate results. The solutions are slightly more accurate than Timoshenko and Gere as more series terms have been used. The buckling stress versus length curve for a plain lipped channel in pure shear has been produced using the reSAFSM with 8 series terms and compared with the signature curve from semi-analytical finte strip method with unrestrained ends (SAFSM). The reSFSM curve has also been compared with the spline finite strip buckling analysis (SFSM) where the ends are fixed against distortion. For the subdivision of the section into 40 strips, the method is accurate to generally better than 1% except in the at L=1600mm where an error of 4% occurs and more series terms would be required at this length before the mode switches to a single distortional buckle at longer half-wavelengths. The buckling stress versus length for a web-stiffened lipped channel with a 5mm swage stiffener has also been investigated. The modes have changed significantly from those of the simple lipped channel. Accuracies generally better than 1% have been achieved compared with the SFSM analysis except at lengths less than 200mm where an error of 1.8% occurs.

ACKNOWLEDGEMENTS

Funding provided by the Australian Research Council Discovery Project Grant DP110103948 has been used to perform this project. The SFSM program used was developed by Gabriele Eccher. The graphics program used to draw the 3D buckling modes was developed by Song Hong Pham under an Australian Government AusAid Scholarship.

REFERENCES

Adany, S. and Schafer, B.W. (2006), “Buckling mode decomposition of thin-walled, single branched open cross-section members via a constrained finite strip method”, Thin-Walled Structures, 64(1): p 12.29.

Anderson, M.S. and Williams, F.W. (1985), “Buckling of simply supported plate assemblies subject to shear loading”, Aspects of the analysis of plate structures, (Eds. Dawe et al), Clarendon Press, Oxford, pp 39-50.



American Iron and Steel Institute (AISI). (2007). “North American Specification for the Design of Cold-Formed Steel Structural Members.” 2007 Edition, AISI S100-2007.

Centre for Advanced Structural Engineering (2006), “THIN-WALL – A computer program for cross-section analysis and finite strip buckling analysis and direct strength design of thin-walled structures”, Version 2.1, School of Civil Engineering, the University of Sydney, Sydney, Australia.

Cheung, Y. K. (1968) “Finite Strip Method Analysis of Elastic Slabs.” ASCE J. Eng. Mech. Div., Vol. 94, EM6, pp. 1365-1378.

Cheung, Y.K. (1976), Finite Strip Method in Structural Analysis, Pergamon Press, Inc. New York, N.Y.

Hancock, G. J. and Pham, C. H. (2011), “A Signature Curve for Cold-Formed Channel Sections in Pure Shear”, Research Report R919, School of Civil Engineering, the University of Sydney, July.

Hancock, G. J. and Pham, C. H. (2012), “Direct Method of Design for Shear of Cold-Formed Channel Sections based on a Shear Signature Curve ”, Proceedings, 21st International Specialty Conference on Cold-Formed Steel Structures (Eds. LaBoube and Yu), St Louis, Missouri, USA, Oct , pp 207 – 221.

Lau, S. C. W. and Hancock, G. J. (1986). “Buckling of Thin Flat-Walled Structures by a Spline Finite Strip Method.” Thin-Walled Structures, Vol. 4, pp 269-294.

Peters, G. and Wilkinson, J.H. (1969), “Eigenvalues of Ax = λBx with Band Symmetric A and B”, Computer Journal, Vol. 12, pp 398-404.

Pham, C. H. and Hancock, G. J. (2009a). “Shear Buckling of Thin-Walled Channel Sections.” Journal of Constructional Steel Research, Vol. 65, No. 3, pp. 578-585.

Pham, C. H. and Hancock, G. J. (2009b). “Shear Buckling of Thin-Walled Channel Sections with Intermediate Web Stiffener.” Proceedings, Sixth International Conference on Advances in Steel Structures, Hong Kong, pp. 417-424.

Pham, C. H. and Hancock, G. J. (2012a). “Direct Strength Design of Cold-Formed C-Sections for Shear and Combined Actions” Journal of Structural Engineering, ASCE, Vol. 138, No. 6, pp 759-768.

Pham, C. H. and Hancock, G. J. (2012b) “Elastic Buckling of Cold-Formed Channel Sections in Shear.” Thin-Walled Structures, Vol. 61, pp. 22-26.

Pham, S.H., Pham, C.H. and Hancock, G.J. (2012a), “Shear Buckling of Thin-Walled Channel Sections with Complex Stiffened Webs”, Research Report R924, School of Civil Engineering, the University of Sydney, January.

Pham, S.H., Pham, C.H. and Hancock, G.J.(2012b), “Shear Buckling of Thin-Walled Channel Sections with Complex Stiffened Webs”, Proceedings, 21st International Specialty Conference on Cold-Formed Steel Structures (Eds. LaBoube and Yu), St Louis, Missouri, USA, Oct 2012, pp 281-296.

Plank, RJ and Wittrick, WH. (1974), "Buckling Under Combined Loading of Thin, Flat-Walled Structures by a Complex Finite Strip Method", International Journal for Numerical Methods in Engineering, Vol. 8, No. 2, pp 323-329.

Standards Australia. (2005). “AS/NZS 4600:2005, Cold-Formed Steel Structures.” Standards Australia/ Standards New Zealand.

Timoshenko, S.P. and Gere, JM. (1961),"Theory of Elastic Stability", McGraw-Hill Book Co. Inc., New York, N.Y.

Turnbull, H.W. (1946), Theory of Equations, Oliver and Boyd, Edinburgh and London.

Wilkinson, J.H. (1958), “The calculation of the eigenvectors of co-diagonal matrices”, Computer Journal, Vol. 1, pp 90-96.

Williams, F.W. and Wittrick, W.H. (1969), “Computational procedures for a matrix analysis f the stability and vibration of thin flat-walled structures in compression”, Int. Journal of Mechanical Sciences, 11, 979-998.

Wittrick, W.H. (1968), “A unified approach to the initial buckling of stiffened panels in compression”, Aeronautical Quarterly, 19, 265-283.



APPENDIX A: BUCKLING STRESS AND SHEAR BUCKLING COEFFICIENT FOR PLAIN LIPPED CHANNEL

Table A. Buckling Stress and Buckling Coefficient for Plain Lipped Channel

Length (mm)

reSAFSM (8 terms) SFSM

Buckling Stress (MPa)

Shear Buckling Coefficient

kv



kv

30 40 50 60 70 80 90

100 120 140 160 180 200 230 270 300 350 400 500 600 700 800 900

1000 1200 1400 1600 1800 2000 2300 2700 3000 3500 4000 5000 6000 7000 8000 9000 10000

3973.952 2281.488 1493.687 1062.087 801.930 639.272 523.503 434.828 322.269 258.758 220.704 196.747 181.059 166.424 155.543 150.447 139.202 132.450 126.762 124.451 122.526 121.494 120.552 120.165 119.323 120.070 122.499 112.839 100.604 85.460 70.262 61.466 49.989 41.132 28.260 19.666 13.830 9.512 6.823 5.059

219.847 126.216 82.834 58.757 44.364 35.366 28.961 24.056 17.829 14.315 12.210 10.884 10.017 9.207 8.605 8.323 7.701 7.327 7.013 6.885 6.778 6.721 6.669 6.648 6.601 6.642 6.777 6.242 5.566 4.728 3.887 3.400 2.765 2.276 1.563 1.088 0.765 0.526 0.377 0.280

3735.470 2188.912 1451.298 1039.176 786.997 628.730 516.653 429.473 318.626 256.026 218.510 194.899 179.450 165.063 154.413 149.455 138.477 131.794 126.202 123.833 121.996 120.920 119.980 119.440 118.620 118.111 117.770 111.764 99.588 84.609 69.605 60.914 49.561 40.795 28.049 19.536 13.786 9.484 6.805 5.056

206.651 121.094 80.288 57.489 43.538 34.782 28.582 23.759 17.627 14.164 12.088 10.782 9.927 9.132 8.542 8.268 7.661 7.291 6.982 6.851 6.749 6.689 6.637 6.608 6.562 6.534 6.515 6.183 5.509 4.681 3.851 3.370 2.742 2.257 1.552 1.081 0.763 0.525 0.376 0.280



APPENDIX B: BUCKLING STRESS AND SHEAR BUCKLING COEFFICIENT FOR STIFFENED WEB CHANNEL

Table B. Buckling Stress and Buckling Coefficient for Stiffened Web Channel

Half-Wavelength/ Length (mm)

reSAFSM (8 terms) SFSM



kv



kv

30 40 50 60 70 80 90

100 120 140 160 180 200 230 270 300 350 400 500 600 700 800 900

1000 1200 1400 1600 1800 2000 2300 2700 3000 3500 4000 5000 6000 7000 8000 9000 10000

4255.165 2583.108 1804.562 1422.517 1212.321 1058.671 894.272 775.562 624.469 539.751 490.584 461.497 444.027 429.122 375.225 341.310 301.747 276.622 251.326 241.917 236.144 217.576 203.801 192.627 172.487 152.111 132.635 115.757 101.839 85.587 69.865 60.935 49.419 40.618 27.918 19.464 13.751 9.461 6.788 5.044

235.404 142.902 99.832 78.696 67.068 58.568 49.473 42.906 34.547 29.860 27.140 25.531 24.564 23.740 20.758 18.882 16.693 15.303 13.904 13.383 13.064 12.037 11.275 10.657 9.542 8.415 7.338 6.404 5.634 4.735 3.865 3.371 2.734 2.247 1.544 1.077 0.761 0.523 0.376 0.279

3963.104 2469.557 1737.175 1374.149 1173.947 1033.547 874.267 758.890 611.755 529.189 481.298 453.027 436.119 421.860 369.918 336.908 298.380 273.891 249.221 240.095 234.431 216.227 202.662 191.610 171.589 151.281 131.884 115.098 101.267 85.120 69.493 60.610 49.148 40.386 27.747 19.339 13.706 9.430 6.766 5.027

219.244 136.619 96.103 76.020 64.944 57.177 48.366 41.983 33.843 29.275 26.626 25.062 24.127 23.338 20.464 18.638 16.507 15.152 13.787 13.282 12.969 11.962 11.212 10.600 9.493 8.369 7.296 6.367 5.602 4.709 3.844 3.353 2.719 2.234 1.535 1.070 0.758 0.522 0.374 0.278



APPENDIX C: FLEXURAL STIFFNESS AND STABILITY MATRICES FOR MTH SERIES TERM

kF CF

T k F CF (C-1) gF CF

T g F ig S CF (C-2) where

CF 1 00 b

0 00 0

3 2b 2 b

3 b2 b

k F =

DX2

DX4

DX4

DX6 2DXY

DX6 D1

DX8

3D12

DX8

D12 2DXY

DX10 D1 2DXY

DX6 D1

DX8

D12 2DXY

DX8

3D12

DX10 D1 2DXY

DX10

2D13 2DY

8DXY3

DX12 D1 3DY 3DXY

DX12 D1 3DY 3DXY

DX14

6D15 6DY

18DXY5

where

DX Dmπ

L A

DYDb A

D1 ν D mπbL A

DXY G t12

mπbL A

A b L

D E t

12 1 ν

g F

ff2

f2

f3

f2

f3

f3

f4 f

f3

f4

f4

f5

f4

f5 f

f5

f6 f

f3

f4

f4

f5 f

f4

f5

f5

f6 f

f5

f6

4f3

f6

f7

6f4

f6

f7

6f4

f7

f8

9f5

where

f mπ

L V2 σ

f mπ

L V2 σ σ

f 1b

V2 σT

V b t L



g S

0 ff 0

f ff3

f2

ff3

ff2

0 f5

f5 0

f mπbL

V2 τ

The stresses σ1, σ2, σT and τ are shown in Fig. 2



APPENDIX D: MEMBRANE STIFFNESS AND STABILITY MATRICES FOR MTH SERIES TERM

kM CM

T k M CM (D-1) gM CM

T g M CM (D-2) where

CM 1 01 0

0 0 1 0

0 1 0 1

0 00 1

k M =

G12

G14

G14

G16

E12

0 G1

2bk

E122

G14bk

E124

0 E122

G12bk

G14bk

E124

E22

E24

E24

G12 bk

E26

where

G1 Gmπ

L V

E1Eb V

E2 E mπ

L V

E12 νEmπbL V

V b t L k

mπL

E E

1 ν

g M

ff2

f2

f3

f2

f3

f3

f4

0 00 0

0 00 0

ff2

f2

f3

f2

f3

f3

f4

where

f mπ

L V2 σ

f mπ

L V2 σ σ

V b t L

Documents

Shear Buckling of Channel Sections with Simply Supported ...steel.org.au/media/File/RR931_Pham_and_Hancock.pdf · school of civil engineering shear buckling of channel sections with